Trigonometry Proof

Date: 7/18/96 at 4:32:59
From: Anonymous
Subject: Trig. Proof
I would appreciate any help that you may be able to give.
(cotX)(cosX) cosX
-------------- = -------
(cotX) + (cosX) 1+ sinX
I have to prove this question, not answer or solve.
Thank you in advance.

Date: 7/18/96 at 13:1:55
From: Doctor Paul
Subject: Re: Trig. Proof
Here's how to prove both sides of your equation are equal:
cot(x) * cos(x) cos(x)
--------------- = ---------
cot(x) + cos(x) 1 + sin(x)
I really didn't know where to begin. There is usually some sort of
substitution to make in these trig identity problems
(like (sin(x))^2 = 1 - (cos(x))^2) but there weren't any obvious
substitutions here. I began by cross multiplying:
[cot(x)*cos(x)]+[cot(x)*cos(x)*sin(x)] = [cot(x)*cos(x)] + [(cos(x)*
cos(x)]
Recall that tan(x) is really sin(x) / cos(x)
and that cot(x) is the 1 / tan(x), so cot(x) = cos(x) / sin(x).
Let's make that substitution:
[cos(x)*cos(x)] [cos(x)*cos(x)*sin(x)]
------------- + --------------------- =
sin(x) sin(x)
[cos(x)*cos(x)] [(cos(x)*cos(x)]
------------- + -------------
sin(x) 1
I don't think it's too hard to see the two sides are equal..just
cancel the two sin(x) terms in the second of the four brackets and
you're in business..
-Doctor Paul, The Math Forum
Check out our web site! http://mathforum.org/dr.math/

Date: 7/18/96 at 15:59:8
From: Doctor Anthony
Subject: Re: Trig. Proof
>prove (cot x * cos x)/(cot x + cos x) = (cos x)/(1 + sin x)
These questions are easily answered if you remember the relationships
between the various trig. ratios. In this case use
cot(x) = cos(x)/sin(x)
So the lefthand side of the above expression can be written:
[cos(x)/sin(x)]*cos(x)
----------------------
cos(x)/sin(x) + cos (x)
Multiplying the top and bottom by sin(x), we get
cos(x)*cos(x)
---------------------
cos(x) + sin(x)*cos(x)
Dividing the top and bottom by cos(x) and we get:
cos(x)
--------- = righthand side.
1 + sin(x)
-Doctor Anthony, The Math Forum
Check out our web site! http://mathforum.org/dr.math/